Integrand size = 14, antiderivative size = 109 \[ \int \frac {(a+a \cos (x))^{3/2}}{x^3} \, dx=-\frac {a \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}}{x^2}-\frac {3}{16} a \sqrt {a+a \cos (x)} \operatorname {CosIntegral}\left (\frac {x}{2}\right ) \sec \left (\frac {x}{2}\right )-\frac {9}{16} a \sqrt {a+a \cos (x)} \operatorname {CosIntegral}\left (\frac {3 x}{2}\right ) \sec \left (\frac {x}{2}\right )+\frac {3 a \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )}{2 x} \] Output:
-a*cos(1/2*x)^2*(a+a*cos(x))^(1/2)/x^2-3/16*a*(a+a*cos(x))^(1/2)*Ci(1/2*x) *sec(1/2*x)-9/16*a*(a+a*cos(x))^(1/2)*Ci(3/2*x)*sec(1/2*x)+3/2*a*cos(1/2*x )*(a+a*cos(x))^(1/2)*sin(1/2*x)/x
Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {(a+a \cos (x))^{3/2}}{x^3} \, dx=-\frac {(a (1+\cos (x)))^{3/2} \left (16+3 x^2 \operatorname {CosIntegral}\left (\frac {x}{2}\right ) \sec ^3\left (\frac {x}{2}\right )+9 x^2 \operatorname {CosIntegral}\left (\frac {3 x}{2}\right ) \sec ^3\left (\frac {x}{2}\right )-24 x \tan \left (\frac {x}{2}\right )\right )}{32 x^2} \] Input:
Integrate[(a + a*Cos[x])^(3/2)/x^3,x]
Output:
-1/32*((a*(1 + Cos[x]))^(3/2)*(16 + 3*x^2*CosIntegral[x/2]*Sec[x/2]^3 + 9* x^2*CosIntegral[(3*x)/2]*Sec[x/2]^3 - 24*x*Tan[x/2]))/x^2
Time = 0.53 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3800, 3042, 3795, 3042, 3783, 3793, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(a \cos (x)+a)^{3/2}}{x^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \sin \left (x+\frac {\pi }{2}\right )+a\right )^{3/2}}{x^3}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \int \frac {\cos ^3\left (\frac {x}{2}\right )}{x^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \int \frac {\sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3}{x^3}dx\) |
\(\Big \downarrow \) 3795 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (-\frac {9}{8} \int \frac {\cos ^3\left (\frac {x}{2}\right )}{x}dx+\frac {3}{4} \int \frac {\cos \left (\frac {x}{2}\right )}{x}dx-\frac {\cos ^3\left (\frac {x}{2}\right )}{2 x^2}+\frac {3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )}{4 x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {3}{4} \int \frac {\sin \left (\frac {x}{2}+\frac {\pi }{2}\right )}{x}dx-\frac {9}{8} \int \frac {\sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3}{x}dx-\frac {\cos ^3\left (\frac {x}{2}\right )}{2 x^2}+\frac {3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )}{4 x}\right )\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (-\frac {9}{8} \int \frac {\sin \left (\frac {x}{2}+\frac {\pi }{2}\right )^3}{x}dx+\frac {3 \operatorname {CosIntegral}\left (\frac {x}{2}\right )}{4}-\frac {\cos ^3\left (\frac {x}{2}\right )}{2 x^2}+\frac {3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )}{4 x}\right )\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (-\frac {9}{8} \int \left (\frac {3 \cos \left (\frac {x}{2}\right )}{4 x}+\frac {\cos \left (\frac {3 x}{2}\right )}{4 x}\right )dx+\frac {3 \operatorname {CosIntegral}\left (\frac {x}{2}\right )}{4}-\frac {\cos ^3\left (\frac {x}{2}\right )}{2 x^2}+\frac {3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )}{4 x}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 a \sec \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \left (\frac {3 \operatorname {CosIntegral}\left (\frac {x}{2}\right )}{4}-\frac {9}{8} \left (\frac {3 \operatorname {CosIntegral}\left (\frac {x}{2}\right )}{4}+\frac {\operatorname {CosIntegral}\left (\frac {3 x}{2}\right )}{4}\right )-\frac {\cos ^3\left (\frac {x}{2}\right )}{2 x^2}+\frac {3 \sin \left (\frac {x}{2}\right ) \cos ^2\left (\frac {x}{2}\right )}{4 x}\right )\) |
Input:
Int[(a + a*Cos[x])^(3/2)/x^3,x]
Output:
2*a*Sqrt[a + a*Cos[x]]*Sec[x/2]*(-1/2*Cos[x/2]^3/x^2 + (3*CosIntegral[x/2] )/4 - (9*((3*CosIntegral[x/2])/4 + CosIntegral[(3*x)/2]/4))/8 + (3*Cos[x/2 ]^2*Sin[x/2])/(4*x))
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) *(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* (m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e /2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n])) Int[(c + d*x)^m*Sin[e/2 + a *(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
\[\int \frac {\left (a +a \cos \left (x \right )\right )^{\frac {3}{2}}}{x^{3}}d x\]
Input:
int((a+a*cos(x))^(3/2)/x^3,x)
Output:
int((a+a*cos(x))^(3/2)/x^3,x)
Exception generated. \[ \int \frac {(a+a \cos (x))^{3/2}}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*cos(x))^(3/2)/x^3,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {(a+a \cos (x))^{3/2}}{x^3} \, dx=\int \frac {\left (a \left (\cos {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}{x^{3}}\, dx \] Input:
integrate((a+a*cos(x))**(3/2)/x**3,x)
Output:
Integral((a*(cos(x) + 1))**(3/2)/x**3, x)
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30 \[ \int \frac {(a+a \cos (x))^{3/2}}{x^3} \, dx=\frac {3}{16} \, \sqrt {2} a^{\frac {3}{2}} {\left (3 \, \Gamma \left (-2, \frac {3}{2} i \, x\right ) + \Gamma \left (-2, \frac {1}{2} i \, x\right ) + \Gamma \left (-2, -\frac {1}{2} i \, x\right ) + 3 \, \Gamma \left (-2, -\frac {3}{2} i \, x\right )\right )} \] Input:
integrate((a+a*cos(x))^(3/2)/x^3,x, algorithm="maxima")
Output:
3/16*sqrt(2)*a^(3/2)*(3*gamma(-2, 3/2*I*x) + gamma(-2, 1/2*I*x) + gamma(-2 , -1/2*I*x) + 3*gamma(-2, -3/2*I*x))
Time = 0.34 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int \frac {(a+a \cos (x))^{3/2}}{x^3} \, dx=-\frac {\sqrt {2} {\left (9 \, a x^{2} \operatorname {Ci}\left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 3 \, a x^{2} \operatorname {Ci}\left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 6 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {3}{2} \, x\right ) - 6 \, a x \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) + 4 \, a \cos \left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 12 \, a \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a}}{16 \, x^{2}} \] Input:
integrate((a+a*cos(x))^(3/2)/x^3,x, algorithm="giac")
Output:
-1/16*sqrt(2)*(9*a*x^2*cos_integral(3/2*x)*sgn(cos(1/2*x)) + 3*a*x^2*cos_i ntegral(1/2*x)*sgn(cos(1/2*x)) - 6*a*x*sgn(cos(1/2*x))*sin(3/2*x) - 6*a*x* sgn(cos(1/2*x))*sin(1/2*x) + 4*a*cos(3/2*x)*sgn(cos(1/2*x)) + 12*a*cos(1/2 *x)*sgn(cos(1/2*x)))*sqrt(a)/x^2
Timed out. \[ \int \frac {(a+a \cos (x))^{3/2}}{x^3} \, dx=\int \frac {{\left (a+a\,\cos \left (x\right )\right )}^{3/2}}{x^3} \,d x \] Input:
int((a + a*cos(x))^(3/2)/x^3,x)
Output:
int((a + a*cos(x))^(3/2)/x^3, x)
\[ \int \frac {(a+a \cos (x))^{3/2}}{x^3} \, dx=\sqrt {a}\, a \left (\int \frac {\sqrt {\cos \left (x \right )+1}}{x^{3}}d x +\int \frac {\sqrt {\cos \left (x \right )+1}\, \cos \left (x \right )}{x^{3}}d x \right ) \] Input:
int((a+a*cos(x))^(3/2)/x^3,x)
Output:
sqrt(a)*a*(int(sqrt(cos(x) + 1)/x**3,x) + int((sqrt(cos(x) + 1)*cos(x))/x* *3,x))